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Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System

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Department of Physics, Tohoku University Joji Nasu

In collaboration with Sumio Ishihara

The 5th Scienceweb GCOE International Symposium

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Strongly correlated electron system

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Charge

Spin Orbital

Lattice

Interplay

Internal degrees of freedom of electron

High-Tc superconductivity Colossal magneto-resistance effect Multi-ferroics Quantum spin liquid etc.

Exotic phenomena

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Orbital Degree of Freedom

3d

Crystalline field with high symmetry Orbital degeneracy

Orbital degree of freedom

(e.g. Perovskite Manganites)

eg orbitals

t2g orbitals

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･Exchange interaction of electronic spins ･Anisotropy of electric conductance ･Coupling with lattice distortion

Anisotropy of electronic distribution

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Orbital Excitation -Orbiton-

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Orbital order

Orbital wave (orbiton)

Spin order

Spin wave(magnon) up down

Spin Degree of freedom

up down

eg orbital degree of freedom

Orbital Pseudo spin

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Example for materials with orbital degree

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LaMnO3: End material of colossal magnetoresistance manganites

KCuF3: the material with one-dimensional magnetism

eg

t2g

Mn3+

eg

t2g

Cu2+

hole

staggered orbital order

Superexchange interaction

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Observation of “Orbiton”

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K. Ishii, et.al., PRB83,241101(R) (2011)

In KCuF3, Resonant Inelastic X-ray Scattering In LaMnO3, Raman scattering

E. Saitoh, et. al., Nature 410 180 (2001)

The collective excitation from orbital order

“Orbiton” has been observed.

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Difference between “Magnon” and “Orbiton”

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Anisotropy of electronic distribution

1) Anisotropy of orbital interaction

2) Strong coupling with lattice distortion

“Orbiton” dispersion is anisotropic.

S. Ishihara and S. Maekawa, PRB62, 2338(2000)

In LaMnO3

Jahn-Teller effect

Orbital degree of freedom

In particular, Dynamical aspect of Jahn-Teller effect might affect orbital dynamics.

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Jahn-Teller Effect Ligand

Energy loss ∝ Q2 Q

Electron-lattice coupling

Lattice potential

Energy gain∝ Q

Lattice must be distorted!

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“Jahn-Teller effect” = Coupling of orbital and lattice distortion

Metal ion

Dynamics of lattice Kinetic energy ∝ ∂2/∂Q2

Orbital dynamics

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Jahn-Teller Effect

Adiabatic potentials

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“Any nonlinear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy”

Jahn-Teller Theorem

Orbital essentially couples with lattice

Low-energy dynamics along rotational direction is expected.

Vibronic Hamiltonian

Kinetic energy of lattice Lattice potential Electron-lattice coupling

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Weak coupling approach

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Vibronic Hamiltonian

Neglected Orbiton-phonon hybridization

Repulsion between orbiton and phonon modes

J. van den Brink, PRL87, 19(2001)

Assumed ground state is unstable.

Energy of orbiton-phonon mode deceases.

Hybridization picture is not correct in strong coupling case.

Orbiton

Phonon

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Purpose

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Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System

On-site dynamical Jahn-Teller effect Inter-site superexchange interaction

Unique excitation due to Jahn-Teller effect

originating from Kinetic energy of lattice vibration

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Model

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:Orbital pseudo-spin operator

eg orbital degree of freedom ＋ 2 kinds of phonons

Exchange interaction

Jahn-Teller coupling is local.

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Symmetry of Hamiltonian

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If , the Hamiltonian is invariant under

and

Existence of Goldstone mode

Symmetry for infinitesimal rotation

Exchange interaction

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Our approach

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Hamiltonian

On site part

Exchange part:

:Off-site part :On-site part

On-site Off-site

: Coordination number

Generalized spin wave approximation with mean-field approximation

Mean-field approximation under Ferro orbital order

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Solution of on-site Hamiltonian

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On-site Off-site

Solve self-consistently

Eigenstates in

with

Exact evaluation of Jahn-Teller coupling

Exact diagonalization by using Householder method ～3000 dimensions ( )

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Inter-site Hamiltonian

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:Projection operator

is expanded by using basis of

Eigenstates of

The states involving ground state are taken into account.

Approximation by bosons

Approximation for orbital operator

F. P. Onufrieva, Sov. Phys. JETP 89, 2270 (1985). N. Papanicolaou, Nucl. Phys. B 305, 367 (1988).

Generalization of spin-wave approximation

Introduction of N～3000 kinds of boson

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Diagonalization of Hamiltonian

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Bosonic Hamiltonian

J. Colpa, Physica A93,327 (1978)

Generalized Bogoliubov transformation

：Bogoliubov boson

Diagonalization for system consisting of kinds of bosons

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Condition

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Ferro orbital order

On one dimensional chain

：Bogoliubov vacuum

Dynamical susceptibility

We can easily generalize its dimension and assumed order.

: There should be Goldstone mode

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Static properties

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• By introducing Jahn-Teller coupling, dynamical JT effect reduces orbital moment. (Ham’s reduction effect)

• In strong coupling limit, dynamical JT effect is proportional to 1/g2.

Non-monotonic behavior

• In all parameter region, the ferro-orbital order is stabilized.

Orbital order is the same regardless of JT coupling

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Weak coupling case

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Gapped excitation

Violation of Goldstone theorem

Gapless excitation: consistent with Goldstone theorem

Orbital excitation mode by our method agrees with that by conventional spin-wave approximation

except for gapless feature.

White lines: Conventional spin-wave approximation where orbiton is hybridized with phonon

J. van den Brink, PRL87, 19(2001)

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Strong coupling case

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Adiabatic planes

Frank-C

on

do

n

excitation

s

Gound state

Exci

ted

sta

tes

Local picture

Phonon sidebands appear around JT energy The flat dispersion Local excitation

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Dispersion of Phonon sidebands

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Phonon sidebands

Large J

Dispersive sidebands

Amplitude is determined by exchange interaction J.

Excitations between adiabatic planes

Propagation of local excitations between adiabatic planes

Extent is determined by JT coupling g.

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Low-energy excitation

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Adiabatic plane

Frank-Condon excitation：Orbital excitation with phonon sidebands

Low-energy excitation：Collective excitation involving lattice vibration

Orbital excitation involving lattice vibration within the lowest adiabatic plane

Frank-Condon excitation

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Low-energy effective model

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Effective model on the lowest adiabatic plane in strong coupling limit (Born-Oppenheimer approximation)

Kinetic energy of rotational mode

Lattice dynamics is strongly coupled with orbital degree of freedom

Born-Oppenheimer approx.: Electronic wave-function

Vibration wave-function

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Low-energy effective model

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Original model Effective model

Good agreement

Low-energy mode corresponds to collective mode of orbital-lattice coupled excitation

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Origin of low-energy excitation

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Degeneracy along rotational direction

Adiabatic plane

Frank-Condon excitation

Orbital excitation involving lattice vibration

Degeneracy along rotational direction

Characteristic of Exe Jahn-Teller system with two kinds of phonons

Adiabatic potential

Orbital excitation involving lattice vibration

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Degree of freedom of phonon

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2 kinds of phonons, Exe JT：

Degree of freedom of phonon Low-energy excitation

1 kind of phonon, Exb1 JT：

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Conclusion

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Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System

Orbital excitation involving lattice vibration

Frank-Condon excitation with phonon sidebands

There are two kinds of excitations

• Dispersive due to superexchange interaction

• Orbital and lattice are strongly coupled.

• This excitation originates from degeneracy of adiabatic plane.